Once again, thank you for the overwhelming response, and I was very happy to see that the two correct answers came from two children – from Muskaan Mittal and Ashish Tutakne – well done! A very creative answer from Pankaj Kakkar as well!
There are many answers to this puzzle, the one I liked the most, I converted to a video, please see attached link:
There are a large number of solutions that utilize the property that you can make triangular shapes with 4 matches (or even more) and make the area to be 4. Ashish Tutakne made one such, which I am including, as are many other variations of that principle.
Pankaj Kakkar from Seattle, US, tried an interesting variation, and while I did not initially understand the answer, I think it is a very creative one and can help create area of all sizes from 0 to 9. I am reproducing the text of his answer directly:
Start with the 3×3 square. Imagine all the sides are rigid lengthwise (I.e., the 3 matches strung together on each side won’t get deformed or bent in any way). Look at the two vertical sides. Keeping one of them fixed in place, slide the other vertical side downward. The two other sides that were horizontal will now slant, and since they are rigid and fixed in length, the vertical sides will come closer. As you continue this process, the shape remains a 4 sided polygon, but the area decreases continuously, starting from 9 and going all the way down to zero, when the two vertical sides will align lengthwise and the slanting sides will also be vertical. Since the change in area is continuous, at some point it will be 4. The calculation of the other properties of the polygon at that point (basically just the slant angle) is left as an exercise for the puzzle master :).
Finally, the most generic answer is to create a star using the twelve matches and adjust the width of the points of the star to get any area between 0 and 11.196, which is the largest area you can get using 12 sticks. Here is the answer:
Hope you all enjoyed the puzzle!